To find the number of points with integral coordinates that lie strictly inside the triangle, we can use the Pick's Theorem.

Pick's Theorem states that for a polygon on a grid with vertices having integral coordinates, the number of points with integral coordinates strictly inside the polygon (denoted as I) can be calculated using the formula:

I = A - B/2 + 1

where A represents the number of points with integral coordinates on the boundary of the polygon, and B represents the number of points with integral coordinates in the interior of the polygon.

To apply Pick's Theorem to the given triangle, we need to calculate A and B.

1. Calculating A (number of boundary points with integral coordinates):

The given triangle has three sides:
- Side 1: (0, 0) to (0, 15)
- Side 2: (0, 15) to (18, 0)
- Side 3: (18, 0) to (0, 0)

The length of side 1 is 15, so there are 15 points with integral coordinates on this side.
The length of side 2 is 18, so there are 18 points with integral coordinates on this side.
The length of side 3 is also 18, so there are 18 points with integral coordinates on this side.

Therefore, the total number of boundary points with integral coordinates (A) is 15 + 18 + 18 = 51.

2. Calculating B (number of interior points with integral coordinates):

To calculate the number of interior points with integral coordinates (B), we can use the Shoelace Formula.

The Shoelace Formula for a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) is given by:

B = (|x1y2 + x2y3 + x3y1 - y1x2 - y2x3 - y3x1| - 2) / 2

Applying the coordinates of the given triangle, we have:

B = (|0*15 + 0*0 + 18*0 - 0*0 - 15*18 - 0*0| - 2) / 2
= (|-270| - 2) / 2
= (270 - 2) / 2
= 268 / 2
= 134

Therefore, the number of interior points with integral coordinates (B) is 134.

3. Calculating I (number of interior points strictly inside the triangle):

Using Pick's Theorem:

I = A - B/2 + 1
= 51 - 134/2 + 1
= 51 - 67 + 1
= -15

Since the number of interior points strictly inside the triangle (I) is -15, which is not a valid answer, the correct option is D - None of these.

Hence, there are no points with integral coordinates strictly inside the given triangle.